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Central Limit Theorem (CLT)
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The Central Limit Theorem (CLT) is a powerful tool of theoretical statistics and probability for understanding the behavior of a random variable. In a nutshell, the CLT states that given a random sample of a population, the probability distribution of the sample mean will approach a normal distribution asymptotically as the sample size increases. This means that the averageness which characterizes the normal distribution, often expressed as “the law of large numbers”, applies to the sample mean regardless of the population’s original distribution.
The CLT is such an important result in probability theory and statistics because it provides a foundation for many other methods. For example, the CLT is a key ingredient in confidence intervals and hypothesis testing, which are the go-to tools for statistical inference. This means that the CLT allows us to use the sample mean to give us an idea of the population mean, and gives us confidence that our sample can accurately inform us about the population.
The CLT is also important to finance, because it helps us to accurately estimate the characteristics of a portfolio of investments. This is especially useful when dealing with large collections of securities, such as mutual funds. By taking a sufficiently large sample of securities and computing the portfolio's mean and standard deviation, we can accurately predict the population characteristics with a greater degree of confidence.
In summary, the Central Limit Theorem is an important and useful law of probability that provides a foundation for understanding how sample means behave. Sample sizes of 30 or more are usually suitable for the CLT to be applicable, and the theorem can be applied in multiple areas, from hypothesis testing to finance. In a nutshell, the CLT proves that the law of large numbers applies to the sample mean regardless of the population’s distribution, and this gives us a greater confidence when looking for the population characteristics.